OBJECTIVE To make semi-log graphs and use them to find models of exponential data.

In Chapter 3 we sketched graphs of exponential functions. But in each case we sketched only a small portion of the graph. This is because exponential functions grow so rapidly that we would need a sheet of paper larger than the pages of this book to sketch them, even for relatively small input values. In this exploration we see how logarithms can be used to help us manage the size of such large graphs. In the process we'll discover how logarithms allow us to "straighten" exponential graphs, helping us to analyze their properties as easily as we analyze lines.

I. How Big Is the Graph?

Let's see what size sheet of paper we need to sketch a graph of$$f(x) = 10^x$$for $x$ between $0$ and $8$.

1. For the function $f$, the input values of $0$ and $8$ correspond to what output values?

   Input value	Output value
   $0$	_______
   $8$	_______

2. Let's make each unit on our graph to be $1$ inch. So we need just $8$ inches for the $x$-axis, which is the width of the pages of this book.
   1. How many inches would we need for the $y$-axis?
   2. To get a better idea of how long the $y$-axis needs to be, convert your answer to part (b) into miles.
   3. Would you be able to see the whole graph at once?
3. What if we decide to let each inch on the $y$-axis represent $1000$ (or $10^3$) units?
   1. How many inches would we need for the $y$-axis?
   2. Convert your answer to part (b) into miles.
   3. Would you be able to see the whole graph at once?
4. Two graphs of $f$ are shown below. Which graph can be used to estimate the values of $f(0.5)? \;f(2.3)? \;f(3.1)? \;f(−0.9)?$
   
   
   
   
   
   
   
   

II. Semi-Log Graphs

None of the graphs we considered in Part I give a satisfactory representation of an exponential function. One way around this dilemma is to use a "logarithmic ruler" or logarithmic scale on the $y$-axis.

When we graph a function and use a logarithmic scale on one of the axes, the resulting graph is called a semi-log graph or semi-log plot. This is equivalent to graphing the points ($x$, $log \;y$).

• To draw a graph of $f$, we plot the points ($x$, $y$).
• To draw a semi-log graph of $f$, we plot the points ($x$, $log \;y$).

Let's sketch a semi-log graph of the exponential function$$y = 100 . 2^x$$for $x$ between $0$ and $10$.

1. Complete the table for the values of $y$ and $log \;y$

   $x$	$y$	$log \;y$		$x$	$y$	$log \;y$
   $0$	$100$	$2$		$6$
   $1$	$200$	$2.3$		$7$
   $2$				$8$
   $3$				$9$
   $4$				$10$
   $5$

2. Draw a semi-log graph of $f$ by plotting the points ($x$, $log \;y$) from the table in Question 1.
   
   
   
   
3. Does the semi-log graph appear to be a line? If so, estimate the slope and the $y$-intercept from the graph.

   Slope: _______

   $y$-intercept: _______

4. Let's show that the graph in Question 2 is a line by finding an algebraic formula for $log \;y$.
   1. Supply the missing reasons.

      $y$	$= 100 . 2^x$	$\color{#00A5DB}{\text{Definition of }y}$
      $log \;y$	$= log(100 . 2^x)$	$\color{#00A5DB}{\text{Take log of each side}}$
      $log \;y$	$= log \;100 + x \; log \;2$	$\color{#00A5DB}{\text{___________________}}$
      $log \;y$	$\approx 2 + 0.3x$	$\color{#00A5DB}{\text{___________________}}$

   2. The last equation in part (a) shows that log y is a linear function of $x$. What is the slope? What is the $y$-intercept? Do your answers agree with the line in the graph you sketched in Question 2?

III. Linearizing Data

A semi-log graph of an exponential function $y = Ca^x$ is a line. We can see this from the following calculations:

$y$	$= Ca^x$	$\color{#00A5DB}{\text{Exponential function}}$
$log \;y$	$= log \;Ca^x$	$\color{#00A5DB}{\text{Take the log of each side}}$
$log \;y$	$= log \;C + x \; log \;a$	$\color{#00A5DB}{\text{Laws of Logarithms}}$

To see that $log \;y$ is a linear function of $x$, let $Y = log \;y$, $M = log \;a$, and $B = log \;C$; then$$Y = B + Mx$$

So if we make a semi-log graph of exponential data, we would get a line with the following properties:

Slope:		$M$	$= log \;a$
$y$-intercept:		$B$	$= log \;C$

Let's see how we can use this information to obtain a function that models exponential data.

1. The following data are exponential. Note that the inputs are not equally spaced.

   $x$	$y$	$log \;y$
   $0$	$10$
   $0.5$	$20$
   $2.0$	$160$
   $3.0$	$640$
   $4.5$	$5120$

   1. Complete the $log \;y$ column in the table.
   2. Make a semi-log graph of the data.
      
      
      
      
2. Let's find a function of the form $y = Ca^x$ that fits the data. (In other words, we need to find $a$ and $C$.)
   1. Estimate the slope and $y$-intercept from the graph.

      Slope:	$M$	= _______
      $y$-intercept:	$B$	= _______

   2. From the above we know that $M = log \;a$ and $B = log \;C$. So

      log $a$	= _______
      log $C$	= _______

   3. Solve the equations in part (b).

      $a$	= _______
      $C$	= _______

   4. So a function that fits the data is$$y = \color{red}{\fbox{  }}.\color{red}{\fbox{  }}^x$$
   5. Check that the values of the function you found in part 2(d) agree with the data.